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Prof. Joseph Fung, FDS
Study on the Volatility Smile of EUR/USD Currency
Options and Trading Strategies
BY
CHEN Duyi
11050098
Finance Concentration
LI Ronggang
11050527
Finance Concentration
An Honors Degree Project Submitted to the School of Business in Partial Fulfillment
of the Graduation Requirement for the Degree of Bachelor of Business
Administration (Honors)
Hong Kong Baptist University
April 2014
Table of Contents
Introduction..................................................................................................................... 2
Statement of the problem ......................................................................................... 2
Literature review....................................................................................................... 3
Objectives of the study.............................................................................................. 4
Data and methodology ..................................................................................................... 5
Data Source .............................................................................................................. 5
Conventions & price calculation................................................................................. 6
Slope ........................................................................................................................ 8
Delta Neutral .......................................................................................................... 12
Trading Strategies........................................................................................................... 13
Strategy 1 ............................................................................................................... 13
Methodology ................................................................................................... 13
Result .............................................................................................................. 14
Before Strategy 2 and Strategy 3 .............................................................................. 15
Strategy 2 ............................................................................................................... 17
Methodology ................................................................................................... 17
Result .............................................................................................................. 19
Strategy 3 ............................................................................................................... 20
Methodology ................................................................................................... 21
Result .............................................................................................................. 22
Reference ...................................................................................................................... 24
Introduction
Statement of the problem
According to Black-Scholes-Merton model, the probability distributions of asset
prices are lognormal. Each option price has an implied volatility. On foreign exchange
markets, the volatility of an exchange rate is not the same at different strike price
and the price of the exchange do not always change smoothly.
An option is priced using volatility depending on its strike price and time to maturity.
A plot of the implied volatility of an option as a function of its strike price is known as
a volatility smile.
The implied volatility of foreign currency options is lower for at-the-money options,
and become higher as the option move to in-the-money or out-of-the-money. In
other words, implied distribution of asset return has heavier tails than the lognormal
distribution as Black-Scholes model suggested which has the same mean and
standard deviation.
Also, the empirical results show that the volatility smiles are not always remaining
constant. What happened if the shapes of the volatility smile changes? What does
the change mean to option? And what is the difference among different maturities?
In this paper, we are going to conduct an empirical study on the volatility surface
through two dimensions. First, we focus on the slope of the volatility smile and the
impact on the option price. Secondly, we test the performance of various trading
strategies concerning the signaling from slope changes, including directional trading,
delta-neutral portfolio and double-traded delta-neutral portfolio. The idea is to try to
utilize the signals from volatility smile and catch profit through corresponding trading
strategies.
Literature review
Unlike equity option market, foreign exchange option market has different
convention of option price and strikes. The dollar price and strikes are not observable
in the market. Instead, it is implied volatility and delta strikes that represent the
option information. Therefore, in the data collecting period, we are facing the
volatility surface and smiles as raw data. Through the “Volatility surfaces: theory,
rules of thumb, and empirical evidence,” (Daglish, T., J. Hull, and W. Suo, 2007) and
“FX Volatility Smile Construction” (Dimitri Reiswich, and Uwe Wystup, 2010), we
understand the construction rules for volatility surface and how we can convert this
convention back to dollar sign which we would be easily to use for strategies testing.
“Jump risk, stock returns, and slope of implied volatility smile, Shu Yan, Moore School
of Business, University of South Carolina, Columbia, SC 29208, United States”
suggested that expected stock return will decrease when stock jumps as empirical
evidence exists for jumps in stock prices. And the jump size can be represented by
the slope of the option implied volatility smile. The theses suggested that the slope
may predict future stock returns. Portfolio with low slope may generate higher
returns than portfolio with high slope. 1.9% monthly profit is generated by buying
the lowest slope portfolio and shorting the highest slope portfolio generates.
“The implied volatility term structure of stock index options , Scott Mixon, Bates
White, LLC, 1300 Eye Street NW, Suite 600, Washington, DC 20005, United States”
suggested that the slope of the term structure of option volatility can forecast future
short dated implied volatility. The hypothesis forecast works better with a volatility
risk premium term. The theses suggested that a portfolio can generate return by
selling volatility to the ones who believe the risk premium interpretation to capture
the gap between implied and realized volatility.
Objectives of the study
Throughout the study, based on the empirical evidence we are dedicated to achieve
three objectives: 1) the general shape of the volatility smiles of EUR/USD currency
options 2) under the volatility surface, how different strategies perform in general.
First, we are going to generate the volatility surface and identify the shape of
volatility smiles of EUR/USD currency options of different time maturities and see
whether they are in U-shape or upward/downward sloping in general, through the
stream of historical options prices. Secondly, we are trying to identify the slope
between two options and explore the signals of change of slope. We would like to
know whether those slopes are changing constantly around an ‘average’ level or has
some obvious trend through different periods. Lastly, we are going to test the trading
strategies of directional trading, delta-neutral portfolio trading and double
delta-neutral portfolio trading based on the volatility slope change. The historical
average return of those strategies based on our volatility smiles and slope changes
will tell us whether we can catch the profit or loss from signals.
Data and methodology
Data Source
We retrieved our data from Bloomberg terminal and other online data providers to
have EUR/USD historical exchange rate, deposit rate for EUR and USD separately,
option implied volatility and implied volatility surfaces for EUR/USD options. The
time span crosses past 10 years from April 1 2004 to April 1 2014. We collected data
for every week namely every 1st, 8th, 15th and 23rd of the month for convenience.
Totally, we have 481 sets of data. One point to address is before 2007, the EUR/USD
option traded in OTC market was of low volume, i.e. low liquidity. Therefore, the data
sets from the early years were not change a lot since there may be less trading at the
time.
For detail, we are using all one-month maturity options, annual deposit rate for USD
and EUR separately from bank average rate of certificate of deposit. The original
call/put strike/premium is quoted in delta/implied volatility convention from
Bloomberg Terminal. After we draw all volatility surfaces , we picked the 25D and
ATM options as they are actively traded most of the time and liquidity risk is low
which is good for our research to avoid the interference from high risk. We also
retrieved historical exchange rate (spot rate) of EUR/USD as we would use it to
calculate the dollar amount of price and strike and the spot is also involved in our
strategy tests.
Conventions & price calculation
The market conventions for options are using delta for strikes and implied volatility
for prices. In our study, our raw data from Bloomberg are all quoted in that
convention and therefore we need calculate the original dollar price and strikes out
of the delta and implied volatility. As the volatility surface from Bloomberg is based
on the Black-Scholes Model, our calculation is literally applying the reverse function
of N(d1) to get the strike prices and then prices are calculated from the model
afterwards.
Black-Scholes Model for call and put:
c = S0 e−rtN(d1) − Ke−qtN(d2)
p = Ke−rt N(−d2) − S0e−qtN(−d1)
d1 =ln (
S0
K ) + (r − q +δ2
2 ) t
δ√t
d2 = d1 − δ√t
The inverse calculation process utilizes the NORMSINV function in Excel , or inverse
function of cumulative normal distribution function to return the true d1 from
cumulative standard normal distribution and the formula for K calculation is as
following:
K(call) = Se−NORMSINV (Δe
(r−q)t)δ√t+(q +δ2
2)t
K(put) = SeNORMSINV(Δe
(r−q)t )δ√t+(q+δ2
2)t
As r is the domestic deposit rate (USD), q is the foreign deposit rate (EURO), δ is the
implied volatility, S for spot rate and t is the time to maturity.
Then using the BS Model, we can easily calculate the option premium out of above
factors.
Volatility Smile
Using 10 years’ monthly average volatility of EUR/USD option which expired in 1
month, we plot the following chart for volatility smile. The skewness is observable
and is due to the interest rate effect of EUR and USD currencies. During the last 10
years’, the interest rate in Europe is higher than that in US most of the times, so the
lowest point is in the right part in the volatility smile.
Slope
We define slope as the difference of volatility between 25D Put and ATM Put divide
by the volatility of ATM Put. ((Vol 25D Put – Vol ATM Put)/Vol ATM Put). In this way,
we set ATM Vol as the benchmark co that we may investigate the relative volatility of
25D Put comparing to ATM Put.
We use put option as the primary data as we can see from the following chart that
implied volatility draw from put option are more of normal distribution than implied
volatility draw from call option.
15D Put EUR 25D Put EUR 35D Put EUR ATM 35D Call EUR 25D Call EUR 15D Call EUR
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Freq
uen
cy
Volatil ity
ATM Vol
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Freq
uen
cy
Volatil ity
35D Call
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Freq
uen
cy
Volatil ity
35D Put
We use 25D Put option volatility in this research. We can see from the following chart
about the slope change in every month that the slope change of 25D options is more
of normal distribution.
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Freq
uen
cy
Volatil ity
25D Call
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Freq
uen
cy
Volatil ity
25D Put
-0.2
5
-0.2
3
-0.2
1
-0.1
9
-0.1
7
-0.1
5
-0.1
3
-0.1
1
-0.0
9
-0.0
7
-0.0
5
-0.0
3
-0.0
1
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
Freq
uen
cy
Slope Change
35D Call Slope Change
-0.2
5
-0.2
3
-0.2
1
-0.1
9
-0.1
7
-0.1
5
-0.1
3
-0.1
1
-0.0
9
-0.0
7
-0.0
5
-0.0
3
-0.0
1
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
Freq
uen
cy
Slope Change
35D Put Slope Change
-0.2
5
-0.2
3
-0.2
1
-0.1
9
-0.1
7
-0.1
5
-0.1
3
-0.1
1
-0.0
9
-0.0
7
-0.0
5
-0.0
3
-0.0
1
0.0
1
0.0
3
0.0
5
0.0
7
0.0
9
0.1
1
0.1
3
0.1
5
0.1
7
0.1
9
Freq
uen
cy
Slope Change
25D Call Slope Change
As volatility change in a certain range, we believe it has the characteristic of mean
reversion.
Delta Neutral
Delta is “The ratio comparing the change in the price of the underlying asset to the
corresponding change in the price of a derivative.” A delta neutral (Delta = 0)
portfolio’ value will not be affected by the underlying assets.
In this case, we trade 25D Put and ATM Put to construct the delta neutral portfolio.
For 25D Put, the delta equals 0.25; for ATM Put, the delta equals 0.5. As a result, if
we would like to long volatility at 25D Put, we may long 2 25 D Put and
simultaneously short 1 ATM Put. Delta of the portfolio equals 2 * 0.25 - 0.5 = 0. On
the contrary, if we would like to short volatility at 25D Put, we may short 2 25 D Put
and simultaneously long 1 ATM Put. Delta of the portfolio equals - 2 * 0.25 + 0.5 = 0.
-0.2
5
-0.2
35
-0.2
2
-0.2
05
-0.1
9
-0.1
75
-0.1
6
-0.1
45
-0.1
3
-0.1
15
-0.1
-0.0
85
-0.0
7
-0.0
55
-0.0
4
-0.0
25
-0.0
1
0.00
5
0.02
0.03
5
0.05
0.06
5
0.08
0.09
5
0.11
0.12
5
0.14
0.15
5
0.17
0.18
5
0.2
Freq
uen
cy
Slope Change
25D Put Slope Change
Trading Strategies
Strategy 1
Methodology
First, we use the information from the change of slope in volatility smile to bet on the
spot of EUR/USD rate.
We tried two strategies to bet on the spot of the EUR/USD rate: (1) If the slope of
volatility smile measured by 25D Put Option and ATM Option increased this month,
we short USD spot and long EUR Spot. (2) If the slope of volatility smile measured by
25D Put Option and ATM Option increased this month, we short EUR spot and long
USD Spot.
As volatility is always described as the fear of investor, the price of a security is
predicted to drop if the volatility of that security increase. Such relationship is proved
to be true between S&P500 and VIX. We would like to test the relationship on
EUR/USD to see whether the spot of one currency is going to drop if the volatility of
that currency increases.
The first strategy is profitable if most of the volatility in EUR/USD is contributed by
USD. The second strategy is profitable if most of the volatility in EUR/USD is
contributed by EUR. Neither of the strategy is profitable if the volatility in EUR/USD is
contributed by both USD and EUR.
Result
Both of the strategies do not earn a normal profit. Both strategies will generate a loss
or have a marginal profit.
We believe the reason is that the volatility in EUR/USD is contributed by both USD
and EUR. As a result, trading spot based on the change of volatility smile is a pure
guess with 50% change to win as one never know whether currency contributes
more volatility in the volatility smile next month. Or fundamental analysis may be
needed to better facilitate this trading strategy.
-10.00%
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
2009
/3/1
2009
/6/1
2009
/9/1
2009
/12/
1
2010
/3/1
2010
/6/1
2010
/9/1
2010
/12/
1
2011
/3/1
2011
/6/1
2011
/9/1
2011
/12/
1
2012
/3/1
2012
/6/1
2012
/9/1
2012
/12/
1
2013
/3/1
2013
/6/1
2013
/9/1
2013
/12/
1
Monthly Return of EUR/USD Spot
Above are the Monthly Return of EUR/USD Spot and Slope Change of EUR/USD
Volatility Smile. We can see that there is no obvious relationship in between which
proves our result above.
Before Strategy 2 and Strategy 3
Other than trading spot, we may also trade volatility based on the change in volatility
smile. As there is no volatility index future for EUR/USD, we first assume we may
trade EUR/USD Historical Volatility Index to see whether the change in volatility smile
has the forecasting ability in future volatility.
The following chart is the 5-year EUR/USD Historical Volatility Index.
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2009
.4
2009
.6
2009
.8
2009
.10
2009
.12
2010
.2
2010
.4
2010
.6
2010
.8
2010
.10
2010
.12
2011
.2
2011
.4
2011
.6
2011
.8
2011
.10
2011
.12
2012
.2
2012
.4
2012
.6
2012
.8
2012
.10
2012
.12
2013
.2
2013
.4
2013
.6
2013
.8
2013
.10
2013
.12
2014
.2
Slope Change of EUR/USD Volatility Smile
If we long the index if the slope change is larger than 0.03 and short the index if the
slope change is smaller than -0.03 at the beginning of every month, the monthly
profits and compounding profits is shown in the charts below.
0
5
10
15
20
25
30
EUR/USD Historical Volatility Index
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
20
09
/4/1
20
09
/7/1
2009
/10/
1
20
10
/1/1
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10
/4/1
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/7/1
2010
/10/
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/1/1
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11
/4/1
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11
/7/1
2011
/10/
1
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/1/1
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/4/1
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/7/1
2012
/10/
1
20
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/1/1
20
13
/4/1
20
13
/7/1
2013
/10/
1
20
14
/1/1
Monthly Profit
Under the assumption that we can trade EUR/USD Historical Volatility Index, i t shows
that the trading strategy based on mean reversion is profitable. It generates 96.41%
return for 10 years which is equivalent to 14.45% annual yield.
After proving trading the change of slope of volatility is profitable, we need to find
out how to realize the strategy by using the financial products we may trade, for
instance, foreign currency options.
Strategy 2
Methodology
In strategy two, we are testing the delta-neutral portfolio that catches the profit from
abnormal deviation from average slope. The idea is based on the mean reversion
process of implied volatility slope between two options with different strikes, in our
case, ATM put and 25D put.
-60.00%
-40.00%
-20.00%
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
120.00%
20
09
/4/1
20
09
/7/1
2009
/10/
1
20
10
/1/1
20
10
/4/1
20
10
/7/1
2010
/10/
1
20
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/1/1
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11
/4/1
20
11
/7/1
2011
/10/
1
20
12
/1/1
20
12
/4/1
20
12
/7/1
2012
/10/
1
20
13
/1/1
20
13
/4/1
20
13
/7/1
2013
/10/
1
20
14
/1/1
Compounding Profit
First is the delta-neutral portfolio construction with two long positions of 25D put
and one short position of ATM put. Secondly, the trigger of trading is the change of
slope of implied volatility between the options. According to mean reversion idea
applied broadly in financial world, we believe whenever the slope becomes larger,
we expect it to become smaller, and back to normal in the following period. Then we
short the portfolio in order to catch the trend. In our research we tested in strategy
two that the change of slope will be recovered immediately in the period (one month)
after and we trade only one time per change no matter how big it changes. And it is
not influenced by the change of underlying asset, i.e. spot price. Similarly, if the slope
becomes far smaller than average, we trade oppositely with long position.
The cost of the portfolio is calculated as following: 1) long position, we calculate the
premium paid directly as cost of strategy. 2) Short position, we follow the market
convention of margin requirement to calculate the delta margin and vega margin.
And we use the margin required as the cost. For the delta margin, since we use the
delta-neutral portfolio, the delta margin requirement is zero at trading time. For vega
margin, we calculate the vega exposure first from the premium, strikes and spot price.
Then we use the 11% margin requirement convention for one-month maturity option.
In our strategy three, we use the same convention to calculate the cost.
Margin Requirement = Delta Margin + 100 × Vega Exposure × IV × 11%
Vega:
Delta:
Then we repeat the strategy every week in the past ten years for 1-month maturity
EUR/USD options. And calculate the return and cumulative return sequentially. The
result is showed in the next part of this paper.
Result
We tested the strategy 2 to our ten year time span with weekly traded option
portfolio of one-month maturity. In the result, we found that only from the year 2007,
trading executions appeared frequently and compounding rate of return of 10 years
is only 88.76%, or 6.56% annually. From Jan.1.2007 to Jan.1.2014, in seven years’
time, the cumulative rate of return is 119.27% or annually 11.87%.
As we analysis further into the result, we found:
1. The reason behind the fact that there is no trading execution before 2007 is that
at the time, people didn’t trade much on EUR/USD option. The volume was low
and liquidity is much lower, especially for the OTC market.
2. The rate of return is not as high as our expected. The reason is that the OTC
market now requires substantial amount of margin. The high vega margin
requirement actually drag down our overall performance. If we ignore the 11%
vega margin with pure profit and loss, the internal rate of return through ten
years is 74.37% per annum.
3. The scale of slope change changes significantly before the crisis, during the crisis
and post crisis. When we tested the trigger of execution, we found that the slope
change appeared to be very large during 2007 and 2008 and the slope barely
changes before 2007 due to low volume. However after 2008, the volume of
trading is high, but the slope changes little from month to month. And as a result,
how we set the trigger, i.e. the ‘significant’ slope change, makes a difference for
our trading performance.
4. Normality of returns. When applying the normality test, four moments are 0.201%
mean, 3.78% variance, 1.62 skewness and 14.90 kurtosis. It means that the
distribution of returns is skewed to the right with fatter tails and less risky of
extreme values. One point to mention is that this distribution shows the return of
our strategy including the periods that we choose not to trade.
Strategy 3
Methodology
In strategy 3, we follow most of the steps in strategy 2 except we will trade more if
the slope of volatility smile changes more. We still holds delta-neutral portfolio and
trade based on the mean reversion process of implied volatility slope.
We tested in strategy three that the change of slope will be recovered immediately
in the period (one month) after and we trade only one time per change no matter
how big it changes. Then we repeat the strategy every week in the past ten years for
1-month maturity EUR/USD options. And calculate the return and cumulative return
sequentially. The result is showed in the next part of this paper.
In strategy 2, we short 2 25D Put Option and long 1 ATM Option when the slope
increase; we long 2 25D Put Option and short 1 ATM Option when the slope
decrease.
In strategy 3, we will require tougher condition to trigger the trade. In strategy 2, we
trade no matter how big it changes while in strategy 3, we trade only when the
change of slope is larger than 0.02. In strategy 3, we trade double portion of options
if the change of slope if larger than 0.02. We short 4 25D Put Option and long 2 ATM
Option when the slope increase by more than 0.02; we long 4 25D Put Option and
short 2 ATM Option when the slope decrease by more than 0.02. We short 2 25D Put
Option and long 1 ATM Option when the slope increase by more than 0.01; we long
2 25D Put Option and short 1 ATM Option when the slope decrease by more than
0.01.
We follow the same cost and margin requirement calculation method as in strategy
2.
Result
In strategy 3, in ten year time span with weekly traded option portfolio of one-month
maturity, we found that compounding rate of return of 10 years is 129.54%, or 8.664%
annually. From Jan.1.2007 to Jan.1.2014, in seven years’ time, the cumulative rate of
return is 129.54% or annually 12.604%. During the crisis time which is 2007 to 2011,
the strategy works the better, yielding 21.525% per annum.
As we analysis further into the result, we found:
(1) Same as strategy 2, the reason that there is no trading execution before 2007 is
the low volume and liquidity in OTC market. The rate of return is not high
because of high vega margin requirement. And our strategy has a better
explaining power during the crisis time
(2) When adapting tougher condition to trigger the trades, the return increases. It is
reasonable because it shows that the explaining power is larger when the change
in slope of volatility smile is larger.
Conclusion
This project is a study on the “Volatility Smile of EUR/USD Currency Options and
Trading Strategies”. By adopting mean reversion methodology in EUR/USD currency
option volatility and holding a delta neutral portfolio, we found the strategy
profitable. The strategy works the best during the crisis times from 2007 to 2010
when it gave an annual return of 21.525%. It shows that the strategy has better
explaining power when the option market is more volatile. In other words, the mean
reversion phenomenon is more obvious when the change in volatility in previous
term is larger.
Reference
[1] Shu Yan, Jump risk, stock returns, and slope of implied volatility smile, Moore School of
Business, University of South Carolina, Columbia, SC 29208, United States
[2] Scott Mixon, Bates White, LLC, The implied volatility term structure of stock index options,
1300 Eye Street NW, Suite 600, Washington, DC 20005, United States
[3] Daglish, T., J. Hull, and W. Suo, “Volatility surfaces: theory, rules of thumb, and empirical
evidence,” Quantitative Finance, 2007, 7 (5), 507–524.
[4] Dimitri. Reiswich, and Uwe. Wystup, “FX Volatility Smile Construction”, CPQF Working Paper
series, No.20, 2010
